Problem Statement
Out of Boundary Paths
There is an m x n grid with a ball. The ball is initially at the position [startRow, startColumn]. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply at most maxMove moves to the ball.
Given the five integers m, n, maxMove, startRow, startColumn, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it modulo 10^9 + 7.
Examples
Example 1:
Input: m = 2, n = 2, maxMove = 2, startRow = 0, startColumn = 0
Output: 6
Explanation: There are 6 paths moving the ball out of the grid boundary.
From starting position, there are 2 ways to move out in 1 step (moving up or left).
From starting position, there are 4 ways to move out in 2 steps (moving right then up, right then left, down then left, down then up).
Example 2:
Input: m = 1, n = 3, maxMove = 3, startRow = 0, startColumn = 1
Output: 12
Explanation: There are 12 paths moving the ball out of the grid boundary.
Constraints
- 1 <= m, n <= 50
- 0 <= maxMove <= 50
- 0 <= startRow < m
- 0 <= startColumn < n
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming with a 3D DP array.
- The state of the DP array is defined by the current position (row, column) and the number of moves remaining.
- For each state, we need to consider all four possible directions to move the ball.
- We need to handle the modulo operation carefully to avoid integer overflow.
- The base case is when the ball is out of the grid boundary, which contributes 1 to the count.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Out of Boundary Paths
There is an m x n grid with a ball. The ball is initially at the position [startRow, startColumn]. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply at most maxMove moves to the ball.
Given the five integers m, n, maxMove, startRow, startColumn, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it modulo 10^9 + 7.
Examples
Example 1:
There are 6 paths moving the ball out of the grid boundary. From starting position, there are 2 ways to move out in 1 step (moving up or left). From starting position, there are 4 ways to move out in 2 steps (moving right then up, right then left, down then left, down then up).
Example 2:
There are 12 paths moving the ball out of the grid boundary.
Problem Breakdown
Constraints:
- 1 <= m, n <= 50
- 0 <= maxMove <= 50
- 0 <= startRow < m
- 0 <= startColumn < n
Key Insights:
This problem can be solved using dynamic programming with a 3D DP array.
The state of the DP array is defined by the current position (row, column) and the number of moves remaining.
For each state, we need to consider all four possible directions to move the ball.
We need to handle the modulo operation carefully to avoid integer overflow.
The base case is when the ball is out of the grid boundary, which contributes 1 to the count.
Real-World Application
This problem has several practical applications:
Robot Path Planning
Calculating the number of possible paths a robot can take to reach a boundary or exit point.
Game Development
Determining the number of ways a game character can move out of a defined area within a certain number of steps.
Probability Calculations
Computing the probability of a random walker exiting a bounded region within a fixed number of steps.